>On Mar 6, 4:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:>> On Tue, 5 Mar 2013 10:29:07 -0800 (PST), Paul <pepste...@gmail.com>>> wrote:>>>> >I suspect there's a theorem about entire complex functions f which have the property that the absolute value of f(z) tends to infinity as the absolute value of z tends to infinity.>> > What does this theorem say? I don't know of any such functions besides polynomials of degree >= 1. Is it the case that the set of functions which have this property is>> > just the set of polynomials of degree >= 1.>>>> Yes.>>>> Non-elementary proof: Look up the Piicard theorems. This is immediate>> even from the "Little" Picard theorem.>>>> Elementary proof: Let g = 1/f. Since f has only finitely many zeroes,>> g is entire except for finitely many poles. Let R be a rational>> function with the same poles as g, and with the same principal>> part at each pole. Then g - R is an entire function that tends>> to 0 at infinity, so g = R.>>Ok, I see why g-R is entire but not why it tends to 0>at infinity. What am I missing?take R the sum of the same principal part at each pole of 1/f>William Hughes>